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11: 36.4 Bifurcation Sets
Special Cases
Hyperbolic umbilic bifurcation set (codimension three): … Hyperbolic umbilic cusp line (rib): …
§36.4(ii) Visualizations
See accompanying text
Figure 36.4.4: Bifurcation set of hyperbolic umbilic catastrophe. Magnify
12: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
(hyperbolic umbilic).
Canonical Integrals
§36.2(ii) Special Cases
§36.2(iv) Addendum to 36.2(ii) Special Cases
13: 13.8 Asymptotic Approximations for Large Parameters
Special cases are … For more asymptotic expansions for the cases b ± see Temme (2015, §§10.4 and 22.5)where w = arccosh ( 1 + ( 2 a ) - 1 x ) , and β = ( w + sinh w ) / 2 . …For the case b > 1 the transformation (13.2.40) can be used. … uniformly with respect to bounded positive values of x in each case. …
14: 19.10 Relations to Other Functions
arctanh ( x / y ) = x R C ( y 2 , y 2 - x 2 ) ,
arcsinh ( x / y ) = x R C ( y 2 + x 2 , y 2 ) ,
arccosh ( x / y ) = ( x 2 - y 2 ) 1 / 2 R C ( x 2 , y 2 ) .
In each case when y = 1 , the quantity multiplying R C supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …
15: 19.16 Definitions
Just as the elementary function R C ( x , y ) 19.2(iv)) is the degenerate caseAll elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …
§19.16(iii) Various Cases of R - a ( b ; z )
The only cases that are integrals of the third kind are those in which at least one b j is a positive integer. All other elliptic cases are integrals of the second kind. …
16: 14.20 Conical (or Mehler) Functions
14.20.3 Q ^ - 1 2 + i τ - μ ( x ) = π e - τ π sin ( μ π ) sinh ( τ π ) 2 ( cosh 2 ( τ π ) - sin 2 ( μ π ) ) P - 1 2 + i τ - μ ( x ) + π ( e - τ π cos 2 ( μ π ) + sinh ( τ π ) ) 2 ( cosh 2 ( τ π ) - sin 2 ( μ π ) ) P - 1 2 + i τ - μ ( - x ) .
Special cases:
14.20.14 π 0 τ tanh ( τ π ) cosh ( τ π ) P - 1 2 + i τ ( x ) P - 1 2 + i τ ( y ) d τ = 1 y + x .
For the case of purely imaginary order and argument see Dunster (2013). …
17: 13.24 Series
13.24.3 exp ( - 1 2 z ( coth t - 1 t ) ) ( t sinh t ) 1 - 2 μ = s = 0 p s ( μ ) ( z ) ( - t z ) s .
(13.18.8) is a special case of (13.24.1). …
18: Errata
  • Section 4.43

    The first paragraph has been rewritten to correct reported errors. The new version is reproduced here.

    Let p ( 0 ) and q be real constants and

    4.43.1
    A = ( - 4 3 p ) 1 / 2 ,
    B = ( 4 3 p ) 1 / 2 .

    The roots of

    4.43.2 z 3 + p z + q = 0

    are:

    1. (a)

      A sin a , A sin ( a + 2 3 π ) , and A sin ( a + 4 3 π ) , with sin ( 3 a ) = 4 q / A 3 , when 4 p 3 + 27 q 2 0 .

    2. (b)

      A cosh a , A cosh ( a + 2 3 π i ) , and A cosh ( a + 4 3 π i ) , with cosh ( 3 a ) = - 4 q / A 3 , when p < 0 , q < 0 , and 4 p 3 + 27 q 2 > 0 .

    3. (c)

      B sinh a , B sinh ( a + 2 3 π i ) , and B sinh ( a + 4 3 π i ) , with sinh ( 3 a ) = - 4 q / B 3 , when p > 0 .

    Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

    Reported 2014-10-31 by Masataka Urago.

  • 19: 22.10 Maclaurin Series
    22.10.7 sn ( z , k ) = tanh z - k 2 4 ( z - sinh z cosh z ) sech 2 z + O ( k 4 ) ,
    22.10.8 cn ( z , k ) = sech z + k 2 4 ( z - sinh z cosh z ) tanh z sech z + O ( k 4 ) ,
    22.10.9 dn ( z , k ) = sech z + k 2 4 ( z + sinh z cosh z ) tanh z sech z + O ( k 4 ) .
    The radius of convergence is the distance to the origin from the nearest pole in the complex k -plane in the case of (22.10.4)–(22.10.6), or complex k -plane in the case of (22.10.7)–(22.10.9); see §22.17. …
    20: 19.24 Inequalities
    Inequalities for R D ( 0 , y , z ) are included as the case p = z . … For x > 0 , y > 0 , and x y , the complete cases of R F and R G satisfy … Inequalities for R C ( x , y ) and R D ( x , y , z ) are included as special cases (see (19.16.6) and (19.16.5)). … Special cases with a = ± 1 2 are (19.24.8) (because of (19.16.20), (19.16.23)), and …The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …